The BLM realization for the integral form of Uv(glm|n)

Abstract

In this paper, we embed the integral form of the quantum supergroup ${\boldsymbol~U}_\bv(\mathfrak{gl}_{m\,|\,n})$ to the product of a family of integral quantum Schur superalgebras. We show that the image of the embedding is a free $\mathbb{Z}[{\boldsymbol~v},{\boldsymbol~v}^{-1}]$-module by finding the basis explicitly and calculating the fundamental multiplication formulas of these bases. Unlike the non-super case, the fundamental multiplication formula, which is the key step, is more complicated since we have to deal with the case of multiplying the odd root vectors. As a consequence, via the base change, we realize the quantum supergroup at roots of unity as a subalgebra of the product of quantum Schur superalgebras. Thus, we find a new basis of quantum supergroups at odd roots of unity which comes from quantum Schur superalgebras.